
theorem
  for S, T being antisymmetric up-complete non empty reflexive RelStr,
X being Subset of [:S,T:] st X is Open holds proj1 X is Open & proj2 X is Open
proof
  let S, T be antisymmetric up-complete non empty reflexive RelStr, X be
  Subset of [:S,T:] such that
A1: for x being Element of [:S,T:] st x in X ex y being Element of [:S,T
  :] st y in X & y << x;
A2: the carrier of [:S,T:] = [:the carrier of S,the carrier of T:] by
YELLOW_3:def 2;
  hereby
    let s be Element of S;
    assume s in proj1 X;
    then consider t being object such that
A3: [s,t] in X by XTUPLE_0:def 12;
    reconsider t as Element of T by A2,A3,ZFMISC_1:87;
    consider y being Element of [:S,T:] such that
A4: y in X and
A5: y << [s,t] by A1,A3;
    take z = y`1;
A6: y = [y`1,y`2] by A2,MCART_1:21;
    hence z in proj1 X by A4,XTUPLE_0:def 12;
    thus z << s by A5,A6,Th18;
  end;
  let t be Element of T;
  assume t in proj2 X;
  then consider s being object such that
A7: [s,t] in X by XTUPLE_0:def 13;
  reconsider s as Element of S by A2,A7,ZFMISC_1:87;
  consider y being Element of [:S,T:] such that
A8: y in X and
A9: y << [s,t] by A1,A7;
  take z = y`2;
A10: y = [y`1,y`2] by A2,MCART_1:21;
  hence z in proj2 X by A8,XTUPLE_0:def 13;
  thus thesis by A9,A10,Th18;
end;
